If both \(A\)  and \(B\)  are positive definite (semidefinite),
it is well-known that \(A\circ B\)  is positive definite (semidefinite);
see, e.g., [1]}.
Moreover,
the celebrated Oppenheim inequality (see [2]} or [1]}) states that
\( \det (A\circ B) \ge \det A\cdot \prod _{i=1}^n b_{ii} \ge \det (AB).\) 
